Chemical Measurements, Error Analysis, and Data Visualization Study Guide

Four Essential Elements of a Chemical Measurement

• Number

  • Represents the quantitative magnitude of the measurement.

  • Obtaining an accurate number requires appropriate control procedures and thorough knowledge of how to perform required mathematical calculations correctly.

• Context

  • Defines the precise identity and nature of what is being measured.

  • Without clear context, reported measurements are easily misinterpreted.

  • Example: In a laboratory practical, context determines whether a standard of $1.6\,\text{grams}$ of oxalic acid refers to $1.6\,\text{grams}$ of pure oxalic acid powder, $1.6\,\text{grams}$ of a powdered mixture containing oxalic acid, or a container bag and powder collectively weighing $1.6\,\text{grams}$.

• Uncertainty

  • Reflects the experimental margin of doubt associated with every measurement, whether explicitly stated or implied.

  • Most commonly expressed via the appropriate use of significant figures.

  • Frequently reported explicitly using statistical parameters such as standard deviation or confidence limits (e.g., $12.7 \pm 0.2\,\text{g}$).

• Units

  • Specifies the scale or dimension of the measurement and must accompany every reported value.

  • Omitting or confusing units drastically changes meaning (e.g., a critical distinction exists between $12.4\,\text{mm}$ and $12.4\,\text{miles}$).

  • When reporting percentages, the specific type of percentage must be explicitly stated, such as:

    • $\% \text{ error}$

    • $\%\,\text{(v/v) ethanol}$

    • $\%\,\text{(w/v) ethanol}$

    • $\%\,\text{(w/v) water}$

Expressing Uncertainty and Significant Figures

• Definition and Scientific Convention

  • The number of digits used to display a numerical value explicitly communicates the level of uncertainty in the underlying measurement.

  • By scientific convention, a properly reported measurement includes all of the certain (known) digits plus the first uncertain (estimated) digit.

  • Comparative Example:

    • Reporting a box width as $14\,\text{inches}$ conveys two significant figures. The digit $1$ is known exactly, while the digit $4$ is uncertain. This implies the actual width lies somewhere between $13\,\text{inches}$ and $15\,\text{inches}$.

    • Reporting a box width as $13.867\,\text{inches}$ conveys five significant figures. The digits $13.86$ are known exactly, while the final digit $7$ is uncertain. This implies the actual width lies between $13.866\,\text{inches}$ and $13.868\,\text{inches}$.

    • Reporting $13.867\,\text{inches}$ when the measurement is only known within $13\,\text{inches}$ to $15\,\text{inches}$ is incorrect. Conversely, reporting $14\,\text{inches}$ when the measurement is known within $13.866\,\text{inches}$ to $13.868\,\text{inches}$ is also incorrect.

• Rules for Counting Significant Figures

  • Include all non-zero digits.

  • Include interior zeros (zeros located between two non-zero digits).

  • Exclude leading zeros (zeros located to the left of the first non-zero digit).

  • Include trailing zeros (zeros at the end of a number) only if a decimal point is explicitly present.

  • Exclude all digits contained within the exponent term of scientific notation (e.g., the digits in $\times 10^{x}$).

• Examples of Significant Figure Counting

  • $0.0102$ contains 3 significant figures.

  • $0.162$ contains 3 significant figures.

  • $120$ contains 2 significant figures.

  • $120.0$ contains 4 significant figures.

  • $1.6 \times 10^{15}$ contains 2 significant figures.

  • $100.2$ contains 4 significant figures.

Reading Scales and Measuring Devices

• Linear Scale Reading Rules

  • When reading continuous linear scales (such as rulers or volumetric glassware), record all marked divisions that are known with certainty.

  • Estimate one final digit beyond the smallest marked division, typically corresponding to one-tenth of that division.

  • Example: If a bar falls between $15.4\,\text{cm}$ and $15.5\,\text{cm}$ on a scale marked in tenths of a centimeter, the measurement is recorded as $15.46\,\text{cm}$. The final digit ($6$) represents an estimated value where minor observer variations may occur.

• Digital Display Reading Rules

  • Record all digits visible on the digital display screen.

  • If the final digit or two fluctuate during observation, record a representative middle value for the first unstable digit.

• Scale Reading vs. Overall Measurement Uncertainty

  • The uncertainty associated with reading a scale represents only one component of the overall measurement uncertainty.

  • Total experimental uncertainty incorporates multiple distinct sources beyond visual scale resolution.

Quality Parameters of Measurement Tools: Accuracy, Precision, and Trueness

• Measurement Tool Quality

  • Different tools used for measuring the same physical property (e.g., measuring a battery diameter using a $\$1$ plastic ruler versus a $\$100$ to $\$1000+$ traceable digital caliper) offer different levels of consistency and reliability.

  • Key parameters describing tool and technique quality include: accuracy, specificity, selectivity, sensitivity, ruggedness, and robustness.

• Accuracy

  • Accuracy represents the primary parameter emphasized in CHEM 3A.

  • Defined as a measure of the total experimental error associated with a measurement.

  • To evaluate accuracy completely, it must be subdivided into two components: precision and trueness.

• Precision

  • Defined (ISO 5725-1:1994en 0.1) as the closeness of agreement between repeated test results obtained from the same object or property under identical conditions.

  • Describes the amount of variance or dispersion across replicate measurements. More precise tools yield tightly grouped data points.

  • Comparative Example: Measuring a battery with calipers may yield repeated values between $11.49\,\text{mm}$ and $11.52\,\text{mm}$, whereas a plastic ruler yields values between $10\,\text{mm}$ and $13\,\text{mm}$. The calipers exhibit higher precision.

  • Driven by random errors, which vary in both direction and magnitude between successive measurements. The distribution of random errors frequently follows a normal distribution (bell curve).

• Trueness

  • Defined (ISO 5725-1:1994en 0.1) as the closeness of agreement between the arithmetic mean of a large series of test results and the true or accepted reference value.

  • Evaluates the presence of systematic (determinate) errors, which consistently displace measurement results in a single direction.

  • Requires comparing experimental measurements against established reference values, ideally using an average of replicate experiments to isolate systematic bias from precision variability.

  • Primary Sources of Reference Values:

    1. Testing reference standards prepared within the experiment (e.g., standard lead solutions prepared from known masses of lead (II) nitrate standard, or certified standard reference materials).

    2. Accepted values published in scientific literature (e.g., comparing calculated atomic weights against International Union of Pure and Applied Chemistry [IUPAC] published standards).

    3. Parallel testing of samples using a second, independent, accepted measurement technique. (Caution: Parallel agreement between two methods does not independently guarantee that either method is true or accurate).

Figures of Merit for Precision and Trueness

• Figures of Merit for Precision

  • Sample Standard Deviation ($s_x$ or $\sigma_{n-1}$): Calculated from repeated measurements of the same sample. A minimum of 3 replicate measurements is required as an absolute lower bound, though some scientific disciplines require up to 21 replicate measurements to evaluate technique precision properly.

  • Relative Standard Deviation (RSD): Calculated directly from the sample standard deviation and the arithmetic mean of the measurements.

  • Calculation Methods: Standard deviations can be computed using scientific calculator statistical functions or spreadsheet commands such as stdev() in Microsoft Excel or Google Sheets.

  • Effect of Averaging: Taking the arithmetic mean of replicate measurements minimizes the impact of random error, thereby enhancing overall experimental accuracy. A sample size of 3 to 5 measurements is often sufficient for precise techniques, but larger sample sizes are necessary when handling high standard deviations.

• Figures of Merit for Trueness

  • Absolute Error ($e$): Retains the same measurement units as the experimental data.

  • Percent Relative Error ($\% e$): A unitless dimensionless ratio.

  • Mathematical Sign: Order of subtraction matters. If an experimental value is lower than the reference value, both the absolute error and percent relative error must be reported as negative numbers.

  • Single vs. Replicate Evaluation:

    • Absolute error and percent relative error calculated on a single measurement reflect overall method accuracy rather than trueness.

    • Replicate measurements are strictly necessary to separate the individual quantitative contributions of precision and trueness.

Sources of Experimental Error

• 1. Equipment Errors

  • Stem from physical limits of measuring hardware and visual scale resolution limits of a trained observer.

  • Example: A ruler with millimeter markings introduces an inherent limitation of $\pm 0.1\,\text{mm}$ in any measurement, requiring higher-precision devices like micrometers to reduce error.

  • Equipment errors are predominantly random, but become systematic if equipment is uncalibrated or improperly calibrated.

• 2. Method Errors

  • Inherent to the procedural design and operational steps used to make a measurement.

  • Typically consist of reproducible systematic errors resulting from how equipment is utilized.

  • Examples:

    • Measuring water volume in open containers introduces systematic loss due to liquid evaporation.

    • Transferring powder from a ziplock bag to weigh paper introduces negative bias from residual powder left inside the bag.

    • Measuring a beach ball's diameter using a straight ruler introduces edge-alignment uncertainties.

    • Determining a beach ball's diameter from circumference measurements introduces systematic error if assuming a perfect sphere.

• 3. Sample Errors

  • Arise from heterogeneity, variability, or temporal instability within the sample being measured.

  • Examples:

    • Measuring individual human body weights yields high variability across distinct individuals.

    • The mass of a marijuana sample shifts over time due to ambient moisture adsorption or drying caused by storage humidity fluctuations.

• 4. Technique Errors

  • Operator-induced errors resulting from poor execution, carelessness, or training gaps.

  • Examples: Blunders, goofs, transposing numbers, omitting digits, recording incorrect units, measuring the wrong parameter, failing to tare a balance, or misreading scale divisions.

  • Minimized through rigorous training and deliberate attention to detail.

• 5. Context Errors

  • Stem from ambiguous measurement parameters or failing to define precisely what is being measured.

  • Example: Measuring the distance around bases in a baseball stadium requires specifying whether measurements are taken along inside corners, center midpoints, outer edges, or runner paths, as well as specifying the league level (Little League, high school, college, or professional).

  • Proper Context Reporting Standard: "The distance around the outside edges of the base pads at Biden Stadium on June 8, 2009 was measured as $360.0\,\text{feet}$."

Principles of Data Visualization and Graphing

• Scatter Plots

  • Designed to evaluate the relationship between two continuous (scalar) variables and illustrate overall trends.

  • Example: Data from the US Geological Survey Friant Dam Monitoring Station plotting river water temperature against time on July 1, 2018 demonstrates a daily temperature minimum near 6:00 AM and a maximum near 4:00 PM.

  • Software Formatting Requirements:

    • Select "Scatter Plot" rather than "Line Plot" in Microsoft Excel or Google Sheets. Line plots treat horizontal x-data as discrete text labels, leading to incorrect numerical spacing.

    • Display discrete experimental data points strictly as standalone markers without connecting lines.

    • Display theoretical models or trendlines strictly as continuous lines without point markers.

    • Exception: High-density continuous data logs (e.g., automated temperature tracking or UV-Vis absorbance spectra across wavelengths) should be displayed as continuous lines without individual data markers.

• Bar Graphs

  • Plot a continuous (scalar) variable on the vertical y-axis (dependent variable) against a discrete or categorical variable on the horizontal x-axis (independent variable).

  • Example: Plotting chemical concentrations collected at discrete water sampling locations.

  • Selection Criterion: If an independent variable is continuous (such as well depth), a bar graph emphasizes differences between specific discrete options, whereas a scatter plot highlights continuous trends across depths.

• Axis Titles, Units, and Legends

  • Every axis displaying scalar data must include a descriptive variable title followed by units enclosed in parentheses, such as: Water Temperature (°C).

  • Unit Omission Exceptions: Units may be omitted only when axes represent arbitrary scales or unitless values (e.g., absorbance or percent error).

  • Legend Requirements:

    • A legend must be included whenever a graph displays two or more dependent variables (e.g., distinguishing blue bars representing calcium ion concentration from orange bars representing magnesium ion concentration).

    • A legend is required if distinct line plots represent separate experimental trials or datasets.

    • A legend is not required for single scatter plots displaying data points alongside one best-fit trendline.

• Visual Design and Formatting Guidance

  • Avoid using red and green combinations together to maintain accessibility for color-blind readers.

  • Ensure high contrast for grayscale or black-and-white printing.

  • Utilize distinct data point shapes and line styles (e.g., solid, dotted, dashed) to differentiate data series.

  • Annotations (such as text boxes, fit equations, arrows, or reference lines) can be added directly or overlaid in presentation tools after exporting charts with transparent backgrounds.

Checklists for Graph Construction

• Bar Graph Checklist

  • Is a bar graph appropriate for the data structure and the conclusions drawn?

  • Are the independent variables spaced evenly on the horizontal axis with appropriate descriptive labels?

  • Is the dependent variable plotted on the vertical axis with a clearly labeled scale?

  • Are both axes labeled with correct variable names and units in parentheses $()$ where applicable?

  • Are the axes and scales set up so the graph utilizes the allotted chart space effectively?

  • Is there a descriptive title or appropriate figure caption?

  • If more than one bar is plotted per horizontal category, does the graph include a legend or clear caption description?

  • Are colors or textures used effectively for visual contrast, printability (grayscale), and color-blind accessibility?

  • Are consistent and appropriate font types and sizes used such that all text is legible from a distance of four feet?

• Scatter Plot or Line Graph Checklist

  • Is a scatter plot or line graph appropriate for the data and the conclusions drawn?

  • Is the independent variable plotted on the horizontal axis with appropriate scale labels?

  • Is the dependent variable plotted on the vertical axis with a clearly labeled scale?

  • Are both axes labeled with correct variable names and units in parentheses $()$ where applicable?

  • Are the axes and scales set up so the graph utilizes the allotted chart space effectively?

  • Is there a descriptive title or appropriate figure caption?

  • If more than one line or data series is plotted:

    • Are distinct point shapes and colors used for different series?

    • Does the graph include an explanatory legend or explicit caption description?

  • Are colors or textures optimized for contrast, print compatibility, and color-blind readability?

  • Are consistent font choices and sizes used to ensure complete legibility from four feet away?

Laboratory Measurement Activity: Seven Station Procedures

• Spreadsheet Data Entry Rules

  • Record individual numerical measurements into the class spreadsheet or data form.

  • Do not enter unit labels directly inside spreadsheet cells; enter pure numbers so software formulas can execute calculations correctly.

  • Use the "increase decimal place" or "decrease decimal place" toolbar formatting buttons in Google Sheets to force the display of required trailing zeros.

• Station Tasks

  • Penny Mass Station: Select one penny and obtain its mass in grams ($\text{g}$), ensuring the balance is tared prior to measurement.

  • Buret Reading Station: Record the volume reading of water in milliliters ($\text{mL}$) from the buret.

  • Beaker Reading Station: Record the volume of water in milliliters ($\text{mL}$) contained in the beaker.

  • Graduated Cylinder Station: Record the volume of water in milliliters ($\text{mL}$) contained in the graduated cylinder.

  • Card Width Station: Measure the width of a credit card or ID card in centimeters ($\text{cm}$).

  • Tennis Ball Station: Measure the diameter of a tennis ball in centimeters ($\text{cm}$).

  • Fidget Spinner Station: Measure the total elapsed time in seconds ($\text{s}$) that a fidget spinner rotates continuously before stopping.